We show that a monic univariate polynomial over a field of characteristic zero, with k distinct nonzero known roots, is determined by precisely k of its proper leading coefficients. Furthermore, we give an explicit, numerically stable algorithm for computing the exact multiplicities of each root over $\mathbb{C}$. We provide a version of the result and accompanying algorithm when the field is not algebraically closed by considering the minimal polynomials of the roots. Then, we demonstrate how these results can be used to obtain the full homogeneous spectra of symmetric tensors—in particular, complete characteristic polynomials of hypergraphs.

中文翻译：

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在给定前导系数的情况下稳定计算已知根的多重性

我们表明，具有特征k的非零已知根的特征零域上的单变量单项多项式是由其适当的前导系数的精确k确定的。此外，我们给出了一种显式的，数值稳定的算法，用于计算每个根的精确的多重性$\mathbb{C}$。通过考虑根的最小多项式，当字段不是代数封闭时，我们提供结果的一种形式和相应的算法。然后，我们演示如何将这些结果用于获得对称张量的全同构谱，尤其是超图的完整特征多项式。